A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

1, 2, 4, 3, 8, 6, 16, 4, 9, 12, 32, 8, 64, 24, 18, 5, 128, 12, 256, 16, 36, 48, 512, 10, 27, 96, 16, 32, 1024, 24, 2048, 6, 72, 192, 54, 15, 4096, 384, 144, 20, 8192, 48, 16384, 64, 32, 768, 32768, 12, 81, 36, 288, 128, 65536, 20, 108, 40, 576, 1536, 131072, 30, 262144, 3072, 64, 7, 216, 96, 524288, 256, 1152, 72, 1048576, 18, 2097152, 6144, 48

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to primorial numbers

Formula

a(n) = A000005(A108951(n)).

a(n) >= A329382(n) >= A329617(n) >= A329378(n).

A020639(a(n)) = A329614(n).

From Antti Karttunen, Jan 14 2020: (Start)

a(A052126(n)) = A329382(n).

a(A002110(n)) = A000142(1+n), for all n >= 0.

a(n) > A056239(n).

a(A329902(n)) = A002183(n).

A000265(a(n)) = A331286(n).

gcd(n,a(n)) = A331283(n).

If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.

A000035(a(n)) = A000035(A000005(n)) = A010052(n).

(End)

Mathematica

Block[{a}, a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Array[DivisorSigma[0, a@ #] &, 75]] (* Michael De Vlieger, Jan 24 2020, after Jean-François Alcover at A108951 *)

Prog

(PARI)
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329605(n) = numdiv(A108951(n));
(PARI) A329605(n) = if(1==n, 1, my(f=factor(n), e=1, m=1); forstep(i=#f~, 1, -1, e += f[i, 2]; m *= e^(primepi(f[i, 1])-if(1==i, 0, primepi(f[i-1, 1])))); (m)); \\ Antti Karttunen, Jan 14 2020
(PARI) A329605(n) = if(1==n, 1, my(f=factor(n), e=0, d); forstep(i=#f~, 1, -1, e += f[i, 2]; d = (primepi(f[i, 1])-if(1==i, 0, primepi(f[i-1, 1]))); f[i, 1] = (e+1); f[i, 2] = d); factorback(f)); \\ Antti Karttunen, Jan 14 2020

Crossrefs

Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).

Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).

Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).

Sequence in context: A243346 A295029 A338918 * A243073 A243345 A297499

Adjacent sequences: A329602 A329603 A329604 * A329606 A329607 A329608

Keyword

nonn

Author

Antti Karttunen, Nov 18 2019

Status

approved